let A be non degenerated commutative Ring; 1. A is not Zero_Divisor of A
assume
1. A is Zero_Divisor of A
; contradiction
then consider b being Element of A such that
A2:
b <> 0. A
and
A3:
(1. A) * b = 0. A
by Def1;
thus
contradiction
by A2, A3; verum